Finite Group Databasebeta

← #108083

Subgroups — #108083

Order 512.

1214 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34212C211yesmaximal, characteristic
2C2×#1892C211yesmaximal, characteristic
3C2×#1520C211yesmaximal, characteristic
4#10846C211yesmaximal, characteristic
5#10721C211yesmaximal, characteristic
6#10714C211yesmaximal, characteristic
7#8023C211yesmaximal, characteristic
8C24×C8C2211yescharacteristic
9C22×C22⋊Q8C2211yescharacteristic
10C22×#103—21no—
11C22×C22⋊C8C2211yescharacteristic
12#2368—22no—
13#2360—22no—
14#2345—22no—
15#2345—21no—
16Q8×C22⋊C4—21no—
17#2329C414yescomplemented
18#2315—21no—
19C4×C22⋊Q8—22no—
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23#2273—21no—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1892—22no—
26#1892C414yescomplemented
27C2×#342—21no—
28C2×#342—21no—
29C2×#340—21no—
30C2×#340—22no—
31C2×#336—22no—
32#1520—22no—
33#1520C414yescomplemented
34C2×#328—22no—
35#1294—22no—
36C23×Q8D411yescharacteristic
37C24×C4C2311yescharacteristic
38C23×C8—21no—
39C23×C8—21no—
40C23×C8—21no—
41C23×C8—21no—
42C23×C8—21no—
43C23×C8—21no—
44C23×C8—21no—
45C23×C8—44no—
46C23×C8C2×C412yes—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8C2×C412yes—
54C2×C22⋊Q8C2×C412yes—
55C2×C22⋊Q8C2×C412yes—
56C2×C22⋊Q8D412yes—
57C2×C22⋊Q8D412yes—
58C2×C22⋊Q8Q812yes—
59C2×C4×Q8—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D412yes—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4D411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42D412yes—
72C2×#103—24no—
73C2×#103—44no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8C2×C412yes—
80#391—22no—
81#391—24no—
82#389—22no—
83#389—24no—
84#387—24no—
85#387—24no—
86#387—24no—
87#385—24no—
88#385—24no—
89#380—22no—
90#380—22no—
91#380—21no—
92#380—21no—
93#380—21no—
94#380—21no—
95#380—21no—
96#380—21no—
97#380—21no—
98#380—21no—
99#380C2×C412yes—
100#380C2×C412yes—
101#380C2×C412yes—
102#380D412yes—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—24no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—24no—
108C4×C22⋊C4—24no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95C2×C412yes—
117#342—42no—
118#342—42no—
119#340—24no—
120#340—44no—
121#336—44no—
122#328—28no—
123C25C2×D411yescharacteristic
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8C22⋊C412yes—
130C22×Q8C22⋊C412yes—
131C22×Q8C4⋊C412yes—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—41no—
138C23×C4—41no—
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×Q811yescharacteristic
151C23×C4#4111yescharacteristic
152C23×C4#4111yescharacteristic
153C23×C4#4111yescharacteristic
154C23×C4#4111yescharacteristic
155C22×C8—24no—
156C22×C8—24no—
157C22×C8—24no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—44no—
163C22×C8—44no—
164C22×C8—44no—
165C22×C8—44no—
166C22×C8—42no—
167C22×C8—42no—
168C22×C8—42no—
169C22×C8—41no—
170C22×C8—41no—
171C22×C8—41no—
172C22×C8—41no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8—41no—
176C22×C8—41no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8C4214yes—
190C22⋊Q8C22⋊C414yes—
191C22⋊Q8C4⋊C414yes—
192C22⋊Q8C4⋊C414yes—
193C4×Q8—82no—
194C4×Q8—82no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C414yes—
220C2×C4⋊C4C4⋊C412yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4SD1612yes—
256C2×C22⋊C4Q1612yes—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—42no—
262C2×C42—42no—
263#103—48no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—41no—
267C22⋊C8—41no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—41no—
279#95—41no—
280C24—42no—
281C24—42no—
282C24C2×C22⋊C411yescharacteristic
283C24C4×D411yescharacteristic
284C24C4×D411yescharacteristic
285C24#12011yescharacteristic
286C24#12011yescharacteristic
287C24#12111yescharacteristic
288C24C22⋊Q811yescharacteristic
289C24C2×SD1611yescharacteristic
290C24C2×Q1611yescharacteristic
291C24#13511yescharacteristic
292C24#13511yescharacteristic
293C24AGL(1,ℤ/8)11yescharacteristic
294C24#13711yescharacteristic
295C24#13711yescharacteristic
296C24#13711yescharacteristic
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—44no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—44no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—82no—
307C2×Q8—81no—
308C2×Q8—81no—
309C2×Q8#9514yes—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4#10314yescomplemented
364C22×C4C2×C4211yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C4⋊C411yescharacteristic
370C22×C4C2×C4⋊C411yescharacteristic
371C22×C4C2×C4⋊C411yescharacteristic
372C22×C4#11711yescharacteristic
373C22×C4#11711yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4C4×D411yescharacteristic
377C22×C4C4×D411yescharacteristic
378C22×C4C4×D411yescharacteristic
379C22×C4C4×D411yescharacteristic
380C22×C4#12011yescharacteristic
381C22×C4#12011yescharacteristic
382C22×C4#12111yescharacteristic
383C22×C4#12111yescharacteristic
384C22×C4#12111yescharacteristic
385C22×C4C22⋊Q811yescharacteristic
386C22×C4C22⋊Q811yescharacteristic
387C22×C4C22⋊Q811yescharacteristic
388C22×C4#12311yescharacteristic
389C22×C4#12311yescharacteristic
390C22×C4#12311yescharacteristic
391C22×C4#12311yescharacteristic
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—42no—
396C2×C8—44no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—44no—
400C2×C8—44no—
401C2×C8—42no—
402C2×C8—44no—
403C2×C8—44no—
404C2×C8—42no—
405C2×C8—82no—
406C2×C8—81no—
407C2×C8—81no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—82no—
422C4⋊C4—81no—
423C4⋊C4—81no—
424C4⋊C4—84no—
425C4⋊C4—84no—
426C4⋊C4#9514yes—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—42no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—81no—
448C22⋊C4—81no—
449C22⋊C4—81no—
450C22⋊C4—81no—
451C42—42no—
452C42—42no—
453C42—82no—
454C42—84no—
455C42—84no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23C4×C22⋊C411yescharacteristic
479C23#37911yescharacteristic
480C23#38011yescharacteristic
481C23#38511yescharacteristic
482C23#38511yescharacteristic
483C23#38611yescharacteristic
484C23#38611yescharacteristic
485C23C2×#10311yescharacteristic
486C23#41611yescharacteristic
487C23#41711yescharacteristic
488C23#41911yescharacteristic
489C23C4×SD1611yescharacteristic
490C23C4×SD1611yescharacteristic
491C23C4×Q1611yescharacteristic
492C23C4×Q1611yescharacteristic
493C23#44011yescharacteristic
494C23#44011yescharacteristic
495C23#44111yescharacteristic
496C23#44111yescharacteristic
497C23#44811yescharacteristic
498C23#44811yescharacteristic
499C23#44911yescharacteristic
500C23#44911yescharacteristic
501C23#45111yescharacteristic
502C23#45111yescharacteristic
503C23#45211yescharacteristic
504C23#45211yescharacteristic
505C23#46011yescharacteristic
506C23#46211yescharacteristic
507C23#46411yescharacteristic
508C23#46411yescharacteristic
509C23#47511yescharacteristic
510C23#47711yescharacteristic
511C23#47911yescharacteristic
512C23#47911yescharacteristic
513Q8—44no—
514Q8—44no—
515Q8—42no—
516Q8—42no—
517Q8—82no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—162no—
579C2×C4—162no—
580C2×C4—162no—
581C2×C4—162no—
582C2×C4—162no—
583C2×C4—162no—
584C2×C4C2×#9511yescharacteristic
585C2×C4C4×C22⋊C411yescharacteristic
586C2×C4#38011yescharacteristic
587C2×C4#38111yescharacteristic
588C2×C4#38511yescharacteristic
589C2×C4#38511yescharacteristic
590C2×C4#38611yescharacteristic
591C2×C4#38611yescharacteristic
592C2×C4B(4,𝔽2)12yes—
593C2×C4#45812yes—
594C8—44no—
595C8—82no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22C22⋊#9511yescharacteristic
637C22C4×#10311yescharacteristic
638C22#176311yescharacteristic
639C22#178711yescharacteristic
640C22#178811yescharacteristic
641C22#179012yes—
642C22#179412yes—
643C22#180311yescharacteristic
644C22#180511yescharacteristic
645C22#186311yescharacteristic
646C22#186311yescharacteristic
647C22#186511yescharacteristic
648C22#186511yescharacteristic
649C22#187811yescharacteristic
650C22#187811yescharacteristic
651C22#188011yescharacteristic
652C22#188011yescharacteristic
653C4—21no—
654C4—21no—
655C4—21no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—44no—
660C4—44no—
661C4—44no—
662C4—82no—
663C4—84no—
664C4—162no—
665C4—162no—
666C4#186712yes—
667C4#188911yescharacteristic
668C4#189011yescharacteristic
669C2—21no—
670C2—21no—
671C2—21no—
672C2—21no—
673C2—21no—
674C2—21no—
675C2—21no—
676C2—21no—
677C2—21no—
678C2—21no—
679C2—21no—
680C2—21no—
681C2#835112yes—
682C2#835212yes—
683C2#1074511yescharacteristic
684C2#1077911yescharacteristic
685C2#1078911yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
17. #2329C4semidirect
26. #1892C4semidirect
33. #1520C4semidirect
363. C22×C4#103semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#342122
C2×#18922
C2×#15202
#108462
#107212
#107142
#80232